Evaluate the determinants to verify the equation

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Evaluate the determinants to verify the equation

Mathematics
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At vero eos et accusamus et iusto odio dignissimos ducimus qui blanditiis praesentium voluptatum deleniti atque corrupti quos dolores et quas molestias excepturi sint occaecati cupiditate non provident, similique sunt in culpa qui officia deserunt mollitia animi, id est laborum et dolorum fuga. Et harum quidem rerum facilis est et expedita distinctio. Nam libero tempore, cum soluta nobis est eligendi optio cumque nihil impedit quo minus id quod maxime placeat facere possimus, omnis voluptas assumenda est, omnis dolor repellendus. Itaque earum rerum hic tenetur a sapiente delectus, ut aut reiciendis voluptatibus maiores alias consequatur aut perferendis doloribus asperiores repellat.

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I have tried solving this and have been unsuccessful. Can anyone help me out?
is this your problem child of a problem?

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lol my problem
ummm the equation after the equal sign. Is that the determinant?
Since i know what the determinant equals but it doesnt equal that
it's asking you to evaluate the determinant and then show it does equal the expression on the RHS (right and side).
well it doesnt LOL
sure? I find the determinant to be \[ bc^3 + ca^3 + ab^3 - (ba^3 + cb^3 + ac^3) \]
ya i got that too
while the RHS, \[ RHS = (-a^2b + a^2c + ab^2 - ac^2 - b^2c + bc^2)(a+b+c) \] \[= ab^3 + bc^3 + ca^3 -(ba^3 - ac^3 - cb^3) \] which is the same.
in the brackets, should be + +
LOL I dont know where I went wrong but when I foiled I Landed up with too many cs
cs?
like the variable c
This problem is training you. Time to move beyond FOIL and multiply things out and keep track of them. Work this again and make sure you can replicate the result.
like what do u mean by moving beyond foil?
I mean you shouldn't need the rubric anymore. And what's more, you are seeing things with more than two terms in each bracket now. You should teach yourself to do that without using foil, a la the following (x + y + z)(a + b + c) = xa + xb + xc + ya + yb + yc + za + zb + zc
ya but like there it was (a-b)(b-c)(c-a)
so first i did (a-b)(b-c) and then i multiplied my answer by (c-a) Is that correct?
exactly so, that's equal to (ab - ac -b^2 + bc)(c-a) = abc - ac^2 -b^2c + bc^2 -a^2b+a^2c + ab^2 - abc = ab^2 + bc^2 + ca^2 - (ac^2 + ba^2 + ac^2)
etc., etc. Lots of ways to skin this cat. But having the discipline to work through it is important.
okkkkk i just caught my mistake LOL
ok, I'm going to move on.
You make one silly mistake and it ruins everything. Thanks James for ur help
Yes. Here's a hint for this problem. Clearly the problem is symmetric in a,b,c. Which means if we swapped the order of them it shouldn't upset the result too much. That being the case, at every step of the way, we should expect a certain symmetry in the terms. So if there's a term such as ab^2 it make sense that we should also expect a term such as bc^2 and ca^2 This is a way to at least sense check your answer at each step.
Alrighty Thanks for ur help :DDDDDDDD

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